The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron

Understand

The study of the moment map from the Grassmannian to the hypersimplex, and the relation between torus orbits and matroid polytopes, dates back to the foundational 1987 work of Gelfand-Goresky-MacPherson-Serganova.

• On the other hand, the amplituhedron is a very new object, defined by Arkani-Hamed-Trnka in connection with scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory.
• In this paper we discover a striking duality between the moment map $\mu:Gr^{\geq0}_{k+1,n}\to\Delta_{k+1,n}$ from the positive Grassmannian $Gr^{\geq0}_{k+1,n}$ to the hypersimplex, and the amplituhedron map $\tilde{Z}:Gr^{\geq0}_{k,n}\to\mathcal{A}_{n,k,2}(Z)$ from $Gr^{\geq0}_{k,n}$ to the $m=2$ amplituhedron.
• We consider the positroid dissections of both objects, which informally, are subdivisions of $\Delta_{k+1,n}$ (respectively, $\mathcal{A}_{n,k,2}(Z)$) into a disjoint union of images of positroid cells of the positive Grassmannian.